U.S. patent application number 12/557123 was filed with the patent office on 2010-03-18 for adaptive signal averaging method which enhances the sensitivity of continuous wave magnetic resonance and other analytical measurements.
This patent application is currently assigned to The Penn State Research Foundation. Invention is credited to Corey Cochrane, Patrick M. Lenahan.
Application Number | 20100066366 12/557123 |
Document ID | / |
Family ID | 41227126 |
Filed Date | 2010-03-18 |
United States Patent
Application |
20100066366 |
Kind Code |
A1 |
Cochrane; Corey ; et
al. |
March 18, 2010 |
Adaptive Signal Averaging Method Which Enhances the Sensitivity of
Continuous Wave Magnetic Resonance and Other Analytical
Measurements
Abstract
This method of adaptive signal averaging is used to enhance the
signal to noise ratio of magnetic resonance and other analytical
measurements which involve repeatable signals partially or
completely obscured by noise in a single measurement at a rate much
faster than that observed with conventional signal averaging. This
technique expedites the signal averaging process because it filters
each individual scan in real time with an adaptive algorithm and
then averages them separately to provide an averaged filtered
signal with less noise. This technique is particularly useful for
any type of continuous wave magnetic resonance experiment or any
other noisy measurement where signal averaging is utilized.
Inventors: |
Cochrane; Corey; (Culver
City, CA) ; Lenahan; Patrick M.; (Boalsburg,
PA) |
Correspondence
Address: |
BUCHANAN INGERSOLL & ROONEY PC
P.O. BOX 1404
ALEXANDRIA
VA
22313-1404
US
|
Assignee: |
The Penn State Research
Foundation
University Park
PA
|
Family ID: |
41227126 |
Appl. No.: |
12/557123 |
Filed: |
September 10, 2009 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61096449 |
Sep 12, 2008 |
|
|
|
Current U.S.
Class: |
324/310 |
Current CPC
Class: |
G01R 33/60 20130101;
G01R 33/56 20130101 |
Class at
Publication: |
324/310 |
International
Class: |
G01R 33/44 20060101
G01R033/44 |
Claims
1. An improved method of signal processing in which a signal
containing repeated measurements is filtered using a signal
processing device to remove noise, wherein the improvement
comprises: filtering the signal with the exponentially weighted
recursive least squares algorithm to generate a set of filtered
signals each signal corresponding to one of the repeated
measurements; and then averaging those signals to generate a signal
that has a significantly higher signal to noise ratio than that of
conventional signal averaging.
2. The method of claim 1 wherein the set of signals is comprised of
continuous wave magnetic resonance measurements.
3. The method of claim 1 wherein the set of signals is comprised of
signals containing physical, chemical, or biological information
and a noise component.
4. The method of claim 1 wherein the signal processing device is a
desktop computer or a laptop computer.
5. The method of claim 1 further comprising averaging each signal
that results from the filtering process of the exponentially
weighted recursive least square adaptive algorithm to provide an
average filtered output corresponding to a signal having reduced
noise.
6. The method of claim 5 also comprising analyzing the signal
having reduced noise to extract physical, chemical, or biological
information.
7. A method of processing a signal comprised of a plurality of
repeated analytic measurements, the signal partially or completely
obscured by noise comprised of the following steps: filtering in a
signal processing device each analytic measurement in the signal in
real time by application of an adaptive algorithm to provide a
filtered output corresponding to each measurement; averaging each
filtered output separately to provide an averaged filtered output
corresponding to each measurement; and combining the average
filtered outputs to create a signal having reduced noise.
8. The method of claim 7 wherein the signal is comprised of
magnetic resonance scans.
9. The method of claim 8 wherein the electronic measurement scans
are EDMR scans.
10. An improved method of determining properties of a material from
continuous wave magnetic resonance scans or other analytical
measurements in which a signal containing repeatable measurements
is partially or completely obscured by noise is processed in a
signal processing device to create a processed signal that is
analyzed to determine properties of the material wherein the
improvement comprises: processing the signal by: filtering each
measurement portion of the signal in real time by application of an
adaptive algorithm to provide a filtered output corresponding to
each measurement; and averaging each filtered output separately to
provide an averaged filtered output to create a signal having
reduced noise.
11. The method of claim 8 wherein the signal having reduced noise
is used to determine at least one of point defects in the material,
free radicals present in the material, identification of selected
atomic scale structure in the material, inorganic substances in the
material, organic substances in the material, and if the material
is a biological material.
12. The method of claim 7 wherein the magnetic resonance signal is
generated to acquire biomedical information.
Description
FIELD OF INVENTION
[0001] The invention relates to signal processing techniques.
BACKGROUND OF THE INVENTION
[0002] Magnetic resonance is an immensely useful analytical
technique that can be applied to electrons, to nuclei, or
sometimes, both simultaneously. Electron spin resonance (ESR) a
technique which is also sometimes referred to as electron
paramagnetic resonance (EPR) and its nuclear analog, nuclear
magnetic resonance (NMR) are among the most powerful and widely
utilized analytical tools of the past sixty years for applications
in medicine, chemistry, biology, solid state electronics,
archaeology, and many other fields, far too numerous to list.
[0003] ESR is applied in areas which are as mundane as evaluating
the shelf life of beer and to areas as exotic as estimating the age
of exceptionally ancient artifacts. ESR is utilized in the
pharmaceutical industry to study the way certain drugs attack
disease and can be utilized to understand the nature of disease at
a fundamental molecular scale. An example of an ailment under study
via ESR is mad cow disease. ESR is used in the electronics industry
to understand fundamental materials based limitations in the
performance of integrated circuits. ESR, in the form of
electrically detected magnetic resonance, may have great potential
in the future in quantum computing.
[0004] Briefly, in electron spin resonance and in other types of
magnetic resonance, energy is absorbed by a spin (that of an
electron in ESR and a nucleus in NMR) when a particular
relationship exists between a large applied magnetic field vector,
the spin center under observation, and the frequency of
electromagnetic radiation (radio frequency or microwave frequency)
applied to the sample under observation. The relationship conveys a
great deal of information about the physical and chemical nature of
the spin's atomic surroundings. Depending upon the specific
application, this information can help evaluate the potential of a
drug in the treatment of disease or identify physical imperfections
that limit the performance of integrated circuits, or determine the
age of an ancient artifact. Many applications are possible.
[0005] Nearly all scientific measurements involve some sort of
electrical signal which encodes useful information. These
electrical signals consist of a component which carries the
physical, chemical, or biological information of interest and a
noise component. Noise is the undesirable component of the total
signal. The ratio of signal to noise is a generally a meaningful
measure of the quality of the scientific measurement. If the signal
to noise ratio falls below a certain value, the measurement becomes
meaningless. The signal to noise ratio is typically a function of
the time involved in making the measurement. When the noise is
random in nature, which is often the case, the signal to noise
ratio can be improved by increasing the time involved in
measurement. This is often done by signal averaging, that is,
repeating a (repeatable) measurement over and over, then averaging
the measurements. In conventional signal averaging, the signal to
noise ratio improves as the square root of the number of
repetitions.
[0006] EDMR typically involves spin dependent recombination (SDR).
EDMR in general and SDR in particular are electron spin resonance
(ESR) techniques in which a spin dependent change in current
provides a very sensitive measurement of paramagnetic defects.
Without special application of digital signal processing
techniques, EDMR measurements involving SDR are about 7 orders of
magnitude more sensitive than conventional ESR. The techniques are
therefore particularly useful in studies of imperfections in the
semiconductor devices utilized in integrated circuits. In such
devices, the dimensions are quite small and can have very low
defect densities. SDR detected EDMR can be utilized in fully
processed devices such as metal oxide semiconductor field effect
transistors (MOSFETs), bipolar junction transistors (BJTs), and
diodes. With some additional improvements, the technique's very
high sensitivity may make it potentially useful for single spin
detection and quantum computing. However, the sensitivity EDMR is
not currently high enough to detect a single spin in the presence
of the noise encountered with present day EDMR spectrometers in a
reasonable amount of time.
[0007] Continuous wave magnetic resonance typically utilizes a
sinusoidal modulation of the applied magnetic field, thereby
encoding the signal in a sinusoid. The amplitude of the modulated
signal is a measure of the magnetic resonance signal, in this case,
an EDMR detected ESR signal. ESR measurements in general and, in
the specific case utilized herein, EDMR, can provide a measure of
the number of paramagnetic defects within the sample under study as
well as the means to identify the physical nature of these defects.
Magnetic resonance in general can provide a very broad range
information about physical and chemical structure. In continuous
wave magnetic resonance, a lock-in amplifier (LIA) is generally
utilized to demodulate the amplitude modulated magnetic resonance
signal to DC, thus exploiting the sensitivity enhancement available
from the phase and frequency detection. This widely used method
effectively attenuates much of the noise in the magnetic resonance
measurement. In the specific EDMR detected ESR example utilized
here, much of the noise is associated with the 1/f noise typically
observed with a DC current produced by the transistor.
[0008] Although lock-in detection is quite powerful, it is often
insufficient to achieve a reasonable signal-to-noise ratio (SNR),
so signal averaging is also often utilized in magnetic resonance.
In cases in which the single measurement SNR is particularly low,
extensive signal averaging may be required to glean useful
information from the magnetic resonance measurements. In our
demonstration we utilize ESR spectra detected through EDMR in
transistors.
[0009] Though work has been performed to remove noise observed in
related fields via software such as nuclear magnetic resonance
(NMR), not much has been done in any area of ESR including
EDMR.
SUMMARY OF THE INVENTION
[0010] We provide a particularly useful signal processing technique
which reduces the noise in all types of continuous wave magnetic
resonance, including EDMR. In addition, the technique can also be
used in other experiments where repetitive measurements are made.
In this technique, we filter each magnetic resonance scan in real
time (EDMR detected ESR in one case) with an adaptive filtering
algorithm to provide a filtered output for each scan. These
filtered outputs are separately averaged to obtain an averaged
filtered output. This updated filtered average can be regressively
used by the adaptive algorithm to continually filter the incoming
magnetic resonance scans. It is precisely this filtered average
which has much less noise to that of the conventional average due
to the reduction of noise in each of the individual filtered
scans.
[0011] The method quite significantly improves the rate of data
acquisition or, essentially equivalently, the signal to noise ratio
of magnetic resonance and other analytical measurements. We
directly demonstrate the power of the method by applying it to
continuous wave magnetic resonance measurements utilizing one
particularly sensitive magnetic resonance method.
[0012] We here disclose a method which greatly improves upon this
standard method of enhancing signal to noise ratios.
[0013] We have demonstrated our invention with one specific type of
continuous wave magnetic resonance, electrically detected magnetic
resonance (EDMR). Although our discussion and our demonstration of
the invention both directly involve EDMR, the invention is equally
applicable to all types of continuous wave magnetic resonance: ESR
(also known as EPR), NMR, Electron Nuclear Double Resonance
(ENDOR), and other magnetic resonance techniques. The technique is
in fact applicable to all types of analytical measurements in which
the ability to acquire a repeatable measurement involving a current
or a voltage is limited by insufficient signal to noise ratios. (It
should thus also be applicable to pulsed magnetic resonance
measurements.) We demonstrate our invention with sensitive EDMR
detected ESR measurements in small devices with relatively low
defect densities. These EDMR measurements are of this type; they
involve repeatable measurements which are limited by low signal to
noise ratios.
[0014] EDMR measurements in small devices provides an excellent
system in which to demonstrate the capabilities of our invention,
which we show can greatly reduce data acquisition time and enhance
signal to noise ratio.
[0015] This filter can be applied to virtually any noisy
measurement to improve signal to noise ratio. This has practical
application in any situation where a noise containing signal is
indicative of a physical or chemical condition, is produced by some
measuring device and results from repeat measurement being taken to
generate multiple signals or scans.
[0016] The technique is performed using a standard personal
computer which is capable of receiving multiple time sampled
signals through an external analog to digital converter. The
software that is used is required to be able to analyze,
manipulate, and store signals. Also, just about any adaptive
algorithm can be used. We choose to use the exponentially weighted
recursive least squares algorithm (EWRLS) because of its superior
performance relative to others. This algorithm is described
below.
[0017] Other features and advantages of our method will become
apparent from a more detailed description of the technique and
through data presented in the drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0018] FIG. 1 is a current noise spectra from a 4H SiC MOSFET
configured in a gated controlled diode biased with three different
voltages.
[0019] FIG. 2 is a block diagram of the adaptive linear
prediction.
[0020] FIG. 3 is a graph showing EDMR amplitude as the magnetic
field increases of individual unfiltered scan (a) compared to the
individual filtered scan (b).
[0021] FIG. 4 is a graph similar to FIG. 3 showing an average of
100 unfiltered scans (a) compared to the average of 85 filtered
scans (b).
[0022] FIG. 5 is a graph similar to FIGS. 3 and 4 showing an
average of 1000 unfiltered scans (a) compared to the average of 985
filtered scans (b).
[0023] FIG. 6 is a graph showing the signal-to-noise ratio (SNR) as
the number of scans increases for both the conventional (a) and
filtered (b) averages.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0024] We provide a real time exponentially weighted recursive
least squares adaptive signal averaging technique which greatly
decreases the amount of time needed for signal averaging of
continuous wave magnetic resonance measurements. The technique
provides a very low cost means to achieve a quite significant
improvement in signal to noise ratio and data acquisition time. We
demonstrate the utility of the technique with very sensitive ESR
measurements using electrically detected magnetic resonance (EDMR)
via spin dependent recombination (SDR) in individual transistors.
However, we emphasize that the invention should be widely
applicable in continuous wave magnetic resonance measurements. In
addition, the method should be useful in enhancing any analytical
measurement in which a repeatable signal is partly or completely
obscured by noise in a single measurement.
[0025] We performed our measurements demonstrating the invention on
4H SiC lateral n-channel MOSFETs. These devices had a gate area of
200.times.200 .mu.m.sup.2 and a thickness of 500 .ANG.. These
devices received a thermal ONO gate growth process. All EDMR
detected ESR measurements were made with the sample at room
temperature and were performed with a fixed gate voltage. All EDMR
spectra reported here were taken with the magnetic field
orientation parallel to the (100) Si/dielectric surface normal.
EDMR measurements were made with a modulation frequency of 1400 Hz
and quite low modulation field amplitude (<0.1 Gauss). The EDMR
measurements were made on a custom built EDMR spectrometer which
utilizes a Resonance Instruments 8330 X-band bridge, TE.sub.102
cavity, and magnetic field controller, a Varian E-line century 4
inch magnet, and power supply. We use a Stanford Research Systems
SR570 current preamplifier to prefilter and amplify the device
currents. We have implemented a virtual lock-in amplifier using
Labview (version 8.2) with the NI PCI 6259 M series DAQ card. This
VLIA is just as good, if not better, than any of the off the shelf
commercial lock-in amplifiers. All software is implemented in
Labview and is run on a Dell Optiplex GX270 desktop computer with a
3.2 GHz processor and 1 GB of RAM.
[0026] Some of the noise sources that are associated with our EDMR
detected ESR measurements include the ambient noise from the
surrounding hardware and, most importantly, the internal shot,
thermal, and flicker noise arising from within the device under
observation.
[0027] FIG. 1 illustrates the current noise spectra from a MOSFETs
configured in a gated controlled diode for three different biasing
conditions. The top plot represents the condition where 0 volts was
applied to the source and drain of the MOSFET which indicates that
spectra observed is the noise that is generated by the preamp. Note
that this is more or less a white spectrum, meaning that the noise
variance at all frequencies is the same. The middle plot represents
the condition where the source and drain of the MOSFET were applied
a small forward bias yielding a dc current of 0.002 .mu.A and the
bottom plot illustrates the condition where the source and drain
were applied a large forward bias yielding a dc current of 5 .mu.A.
The latter configuration corresponds to the biasing condition that
results in maximum recombination and the operating point of our
EDMR experiments. Note that it is this spectrum is significantly
different than the other two. The reason for this is because of the
significant flicker and shot noise that is introduced with larger
dc currents. This indicates that the dominating source of noise in
the EDMR detected ESR measurement is due to flicker and shot noise,
that is, noise coming from the sample under measurement, and that
the noise from the preamp only becomes a problem when smaller
devices (smaller currents) are being used.
[0028] Initially, we attempted to reduce the noise observed in the
EMDR experiments with adaptive noise cancellation techniques with a
field programmable gate array (FPGA) before lock-in detection. The
logic of processing EDMR signals before lock-in detection was in
hopes that a better representation (ie: improved SNR) of the
amplitude modulated input signal would result in an improved SNR
signal at the output of the LIA. It turned out that only minimal
improvement was achieved because, as mentioned earlier, the
majority of the noise in the EDMR experiment arise from the device
under study and not the surrounding ambient noise. Also, lock-in
detection itself is an extremely effective means of removing noise
because it is not only frequency sensitive, but it is sensitive to
phase as well. Therefore, the only noise that contaminates the EDMR
signal is the noise that has frequency content near that of the
modulation frequency. As a result, we decided to move our search to
the output of the LIA for an effective way to enhance the
sensitivity of EDMR. This LIA output signal is a time varying
voltage. In most ESR measurements, including our EDMR detected ESR
measurements, the time variation corresponds to variation in an
applied magnetic field.
[0029] In some cases, the devices under study have very few defects
which make signal acquisition very difficult and time consuming.
These devices provided the opportunity to clearly demonstrate the
power of the method: a way to expedite the averaging process by
utilizing the predictability of the autoregressive noise features
at the output of the LIA. The time constant of the LIA determines
the correlation between successive samples and hence, the
predictability. We term this tool an adaptive signal averager (ASA)
which utilizes adaptive linear prediction as illustrated in FIG. 2.
It works by using the conventional scan average as the desired
response in an adaptive linear prediction configuration. The linear
predictor w.sub.n is a finite impulse response (FIR) filter of
length p and the input to the linear predictor is the tapped
delayed noisy EDMR signal x(n).
w.sub.n=[w.sub.n(1), w.sub.n(2), . . . , w.sub.n(p)].sup.T (1)
x(n)=[x(n-1), x(n-2), . . . , x(n-p)].sup.T (2)
The prediction or estimate d.sub.est(n) of the desired signal is
simply computed by the inner product of these two vectors.
d.sub.est(n)=x.sup.T(n)w.sub.n (3)
The estimate is then subtracted from the scan average to form an
instantaneous error e(n) which is used in an algorithm to update
the weights of the FIR predictor.
e(n)=d(n)-d.sub.est(n) (4)
[0030] There are many forms of adaptive filters but the two most
widely used and efficient are the least mean squares (LMS) and
recursive least squares (RLS) adaptive filters. These filters are
advantageous because they are capable of tracking non-stationary
signals and noise and neither algorithm requires an estimate of the
signal or noise statistics. This is desired for virtually all
continuous wave magnetic resonance experiments because these
statistics vary from sample to sample and may vary over time, over
temperature, over many possible variables possible in the
measurements. This variability in measurement is the case in the
EDMR measurements utilized in demonstrating our method. The main
advantage of the RLS algorithm has over the LMS algorithm is that
it has about an order of magnitude faster convergence time, though,
in most cases, the LMS algorithm is known to have better tracking
performance. Although many adaptive filter algorithms have been
developed and many could be utilized in our discussion, we choose
one, the one which is likely the most advantageous, the RLS
algorithm. However it is our intent in this patent disclosure to
include ALL adaptive algorithms. To increase the tracking
performance of the RLS algorithm, we utilized the exponentially
weighted RLS (EWRLS) algorithm by incorporating an exponentially
weighing factor .lamda. into the system. By doing this, the
algorithm effectively becomes more sensitive to changes in the
noise environment. The exponential weighting factor .lamda.
controls the memory of the system and is chosen to be in the range
0<.lamda.<1. The EWRLS algorithm becomes the RLS algorithm
when .lamda. is chosen to be 1 which provides the system with
infinite memory.
[0031] The EWRLS algorithm attempts to minimize the exponentially
weighted sum of squared errors cost function which is given by
equation (5).
.xi. ( n ) = i = 0 n .lamda. n - i e ( n ) 2 ( 5 ) ##EQU00001##
In order to minimize this cost function, the gradient is taken with
respect to the weights of the FIR predictor and set equal to zero
which is given by equation (6).
.gradient. .xi. ( n ) = - i = 0 n .lamda. n - i x ( i ) e ( i ) = 0
( 6 ) ##EQU00002##
This resultant vector represents the direction of steepest decent
on the sum of squared error surface. Plugging in for the error and
rearranging yields the set of linear equations given in equation
(7).
[ i = 0 n .lamda. n - i x ( i ) x T ( i ) ] w n = i = 0 n .lamda. n
- i d ( i ) x ( i ) ( 7 ) ##EQU00003##
This result can be simplified by realizing that the terms in the
brackets on the left is the summation of exponentially weighted
deterministic autocorrelation matrices R.sub.x(n) of the input
signal from time index 0<i<n and the right hand side is the
summation of exponentially weighted deterministic cross correlation
vectors r.sub.dx(n) of the desired signal and the input signal from
time index 0<i<n. By this realization, equation (7) in matrix
form is equivalent to equation (8).
R.sub.x(n)w.sub.n=r.sub.dx(n) (8)
Therefore, the weight vector w.sub.n is found by multiplying the
cross correlation vector r.sub.dx(n) with the inverse correlation
matrix R.sub.x.sup.-1(n). Calculation of this inverse is
computationally intense so it is not desirable to calculate it
every time a new sample is presented to the system. Therefore, one
way to reduce the computational time is to realize that R.sub.x(n)
and R.sub.x.sup.-1(n) can be solved recursively. It can be easily
shown that,
R.sub.x(n)=.lamda.R.sub.x(n-1)+x(n)x.sup.T(n) (9)
Now that R.sub.x(n) can be solved for in terms of R.sub.x(n-1),
there needs to be a way to compute the inverse of this matrix. This
is called the matrix inversion lemma. The inverse of the
exponentially weighted autocorrelation matrix in equation (9) can
be solved using Woodbury's identity. Woodbury's identity states
that matrix A of equation (10) can be inverted with the relation
shown in equation (11). This identity only holds if A and B are
positive-definite p-by-p matrices, D is a positive-definite n-by-p
matrix, and C is an p-by-n matrix. The relation is easily shown by
computing AA.sup.-1=I, where I is the identity matrix.
A=B.sup.-1+CD.sup.-1C.sup.T (10)
A.sup.-1=B-BC(D+C.sup.TBC).sup.-1C.sup.TB (11)
Note that the following derivation is for real valued data. The
transpose operations would be replaced with the hermitian operator
for imaginary valued data. Comparing equations (10) and (11), it
can be realized that
A=R.sub.x(n) (12)
B.sup.-1=.lamda.R.sub.x(n-1) (13)
C=x(n) (14)
D=1 (15)
Then, plugging equations (12)-(15) into equation (11), the
exponentially weighted inverse autocorrelation matrix can be
computed recursively as follows.
R x - 1 ( n ) = .lamda. - 1 R x - 1 ( n - 1 ) + .lamda. - 2 R x - 1
( n - 1 ) x ( n ) x T ( n ) R x - 1 ( n - 1 ) 1 + .lamda. - 1 x T (
n ) R x - 1 ( n - 1 ) x ( n ) ( 16 ) ##EQU00004##
This equation is usually reduced into simpler form, as shown in
equation (17)
R x - 1 ( n ) = 1 .lamda. [ R x - 1 ( n - 1 ) - g ( n ) z T + ( n )
] where , ( 17 ) z ( n ) = R x - 1 ( n - 1 ) x ( n ) ( 18 ) g ( n )
= 1 .lamda. + x T ( n ) z ( n ) z ( n ) = R x - 1 ( n ) x ( n ) (
19 ) ##EQU00005##
The next step is to solve for the weight update. As stated earlier,
the weight vector is found by multiplying the cross correlation
vector r.sub.dx(n) with the inverse correlation matrix
R.sub.x.sup.-1(n). To reduce computation, r.sub.dx(n) is solved
recursively in a similar fashion to that of R.sub.x(n) and is shown
below.
r.sub.dx(n)=.lamda.r.sub.dx(n-1)+d(n)x(n) (20)
The weight vector is found by computing the product of the
autocorrelation matrix R.sub.x.sup.-1(n) obtained in equation (16)
and the recursive cross correlation vector r.sub.dx(n) formed by
equation (21) and realizing that
R.sub.x.sup.-1(n-1)w.sub.n-1=r.sub.dx(n-1).
w.sub.n=R.sub.x.sup.-1(n)r.sub.dx(n)=w.sub.n-1+g(n).alpha.(n)
(21)
where g(n) was defined previously and .alpha.(n) is the a priori
error. The priori error is the error that occurs when using the
previous set of filter coefficients w.sub.n and is shown below,
.alpha.(n)=d(n)-x.sup.T (n)w.sub.n-1 (22)
It is easy to see that the computation has been reduced
significantly from the conventional LS algorithm because of the
recursive nature of the autocorrelation and cross correlation
functions. R.sub.x.sup.-1(n) can be initialized directly or by
forming the matrix .delta.I, where .delta. is a constant called the
regularization parameter and I is the identity matrix. The
initialization of .delta. depends on the SNR of the signal under
observation and should be calculated with the following
equation:
.delta.=.sigma..sub.u.sup.2(1-.lamda.).sup..alpha. (23)
where .sigma..sub.u.sup.2 represents the noise variance of an
individual EDMR scan, .lamda. is the exponential weighting factor,
and .alpha. is a constant to be determined by the SNR of the EDMR
scan. .alpha. should be chosen to be 1 for SNR>30 dB,
-1<.alpha.<0 for SNR.about.10 dB, .alpha.<-1 for
SNR<-10 dB.
[0032] In some cases, the RLS algorithm can become unstable due to
its mathematical formulation. This occurs when the inverse
autocorrelation matrix loses its symmetry property. This can be
avoided simply by calculating the lower (or upper) triangle of the
inverse autocorrelation matrix and filling the upper (or lower)
triangle to preserve its symmetry property. Not only is this
technique attractive because it prevents instability, but it also
reduces computation. We utilized this method because we initially
encountered instability problems.
[0033] As mentioned earlier, the ASA filters each incoming EDMR
scan in real time via the EWRLS algorithm. The conventional average
is used as the desired signal in the algorithm and can be thought
of as an approximate guide for the filter to follow. Therefore, the
filter allows the noise that it sees to pass, but it effectively
reduces the variance of it, thereby acting as a low pass filter
with a time constant proportional to (1-.lamda.).sup.-1. This is
ideal for magnetic resonance measurements, because one usually
sacrifices a smaller time constant for the observation of smaller
signals. As a result, each individual spectrum will contain more
noise and will require the need for longer signal averaging to
obtain a reasonable SNR. The filtered output scans are then
averaged separately. The underlying idea for this action is that,
because the noise of the filtered scans is reduced, the noise in
the filtered average will be reduced faster than that of the noise
in the conventional average.
[0034] In conventional signal averaging, assuming the noise has a
Gaussian distribution and is independent and identically
distributed (iid) with variance .sigma..sub.u.sup.2, the averaged
noise variance .sigma..sub.uN.sup.2 is reduced by a factor of the
number of scans N in the average as given in equation (14).
.sigma. uN 2 = .sigma. u 2 N ( 24 ) ##EQU00006##
The reduction in noise of the ASA can be determined by analyzing
the error that is introduced into the algorithm. For an individual
scan, the error introduced into the system by the filter is the
combination of the averaged noise in the conventional average
u.sub.N(n) with variance .sigma..sub.uN.sup.2 and the prediction
error of the filter v(n).
e(n)=d(n)-d.sub.est(n)=[d(n)+u.sub.N(n)]-[d(n)+v(n)]=u.sub.N(n)-v(n)
(25)
For ease of analysis, it is assumed that the prediction error is
also Gaussian random variable and has 0 mean and variance
.sigma..sub.v.sup.2. Therefore, the variance of the error
.sigma..sub.e.sup.2 for an individual scan is found by adding the
variances of each of the random variables.
.sigma. e 2 = .sigma. uN 2 + .sigma. v 2 = .sigma. u 2 N + .sigma.
v 2 ( 26 ) ##EQU00007##
If M filtered scans are averaged, then the reduction in noise
variance achieved by the ASA is simply given in equation (17).
.sigma. e M 2 = .sigma. u 2 NM + .sigma. v 2 M ( 27 )
##EQU00008##
where M<N. The reason M scans are averaged and not N is because
we want the conventional average to build up a reasonable desired
response before the filter is applied so a better prediction can be
achieved. N is not that much greater than M so they approximately
equal when considering longer averages. Therefore, as N and M get
larger, the faster the first term in equation (17) dies away which
implies that the dominating source of noise will eventually only be
due to the prediction error of the filter. This is desirable
because it is this first term that actually slightly biases the
ASA. By allowing the conventional average to build a reasonable
desired signal before the filter is applied, the noise bias is
gradually removed. It turns out that not many scans are required to
be averaged for this bias to be removed.
[0035] As discussed earlier, the prediction of the desired signal
is always better than or equal to that of the noisy input because
the filter is optimized to minimize the sum of squared errors.
Therefore, the reduction in noise of the filtered average will
always be better than that of the original average over time,
despite being averaged with fewer scans. As a result, one can see
why this averaging process is expedited; averaging a random
variable with a small variance (prediction error) will converge
much faster than averaging a random variable with larger variance
(noise error).
[0036] The EWRLS ASA was implemented in Labview version 8.2
software and applied to EDMR detected ESR for 4H SiC MOSFETs. The
spectrometer settings used in the scan were purposely chosen to
reduce the SNR of the signal so to better visually observe the
improvement of the filtered signal. The variables that were used in
the EWRLS algorithm were .lamda.=0.98, .delta.=1, p=32 taps, and
the filter was applied after averaging 15 scans. FIG. 3 compares
the performance of the filter of an individual scan. With the
signal amplitude normalize to 1, the noise variance was calculated
to be .sigma..sub.u.sup.2=0.0315 in the unfiltered trace and was
calculated to be .sigma..sub.u.sup.2=0.00278 for the filtered
trace. (These values were calculated by taking the variance of the
difference between the individual scan and the final average.) As a
result, a 11.3 times reduction in noise variance was observed in a
single scan which corresponds to a 11.3 times reduction in time as
well. FIG. 4 compares the average of 100 unfiltered scans and the
average of 85 filtered scans. Note that the filtered average isn't
as noisy as the conventional average and has almost converged to
its final value. FIG. 5 compares the average of 1000 unfiltered
scans and the average of 985 filtered scans. Note that noise is
present in the unfiltered average whereas the noise is not visually
observable in the filtered average. Also, the variance of the noise
that remains in the conventional average after 1000 scans is
approximately equal to the noise variance in the filtered average
after about 90 or so scans as illustrated in FIG. 6. As a result,
the reduction in a noise variance by factor of 11.3 in an
individual scan is equivalent to a reduction in time by the same
amount as illustrated in FIG. 6. In this particular experiment, the
conventional average (1000 scans at 1 minute each) took 1000
minutes to complete. The filtered average converged in
approximately 90 scans which amounts to 910 less minutes of
scanning time to obtain a comparable SNR. This is a very
significant consideration, especially for measurements that require
days of signal averaging. A signal that would usually require 10
days of signal averaging would be reduced to averaging for less
than 1 day (assuming similar filter performance).
[0037] A concern one might have would be when to apply the filter.
It turns out, that even if the SNR is less than 1, the filtered
average will converge to the same result as the original average.
This is acceptable to do so long as a sufficient number of scans
are averaged first before the filter is applied to remove much of
the noise bias, as discussed earlier. We applied the ASA to a 4H
SiC BJT which has smaller amplitude hyperfine structure that is
unobservable until at least 20 or so scans in the average. The
filter was applied after 15 scans (before any of the small
hyperfine structure was observed) and our results show that the
unfiltered average and the filtered average are identical after 250
scans. The only difference is that the filtered average converged
in many fewer scans.
[0038] We have demonstrated that the EWRLS ASA is an extremely
useful and efficient tool for enhancing the rate of data
acquisition and the signal to noise ratios for continuous wave
magnetic resonance through our demonstrations utilizing EDMR
detected ESR. We have shown that the EWRLS ASA method is capable of
reducing the noise variance by a factor of 11.3 in a magnetic
resonance trace and, as expected, the average of the filtered scans
was shown to converge by a similar factor. Our method can be used
in any situation in which a set of noise-containing signals is
generated through measurements repeated over and over again and
then these signals are averaged. The requirement is only that the
signal under study, which may be largely or completely obscured by
noise in a single trace, is repeatable. It should be emphasized
that, although our demonstration utilizes one specific continuous
wave magnetic resonance technique, EDMR, the approach is quite
widely applicable to other measurements: essentially any type of
continuous wave ESR measurement and other analytical measurements
in which repeatable (somewhat to very noisy) signals are
encountered. Thus, in addition to the specific variety of magnetic
resonance measurements utilized in the demonstration, continuous
wave EDMR, the method can be used for other measurements including
MRI scans. This filter is even successful when the SNR of an EDMR
scan is less than 1. With such great reduction in noise, the ASA
effectively expedites the time of averaging.
[0039] Depending upon the circumstances involved in the application
of our method, it may be utilized in many ways which, at the very
least, would save a great deal of time and cost and, at most, could
significantly enhance the power of important analytical tools such
as continuous wave magnetic resonance. This filter can also be
applied to any measurement in which a repeatable measurement is
available and in which extremely high sensitivity and relatively
short acquisition times are required. A few, among many, possible
applications of this invention include quite significant
improvement in measurements which can determine the structure and
density of performance limiting or performance enhancing point
defects present in a semiconductor or insulator materials and
devices, identifying biologically/medically important free radicals
present in tissue, provide information with regard to the
surroundings of selected molecules/free radicals in inorganic and
organic materials, including biological materials. This method may
also be very useful in quantum computing experiments where weeks of
signal averaging may currently be required for near single spin
detection sensitivity.
[0040] One could envision utilization of the invention in
biomedical applications in which the far more rapid data
acquisition which the technique provides may aid a physician in the
treatment of illness or at least reduce the time a patient is
subjected to an unpleasant diagnostic experience. A reduction of a
factor of ten in the amount of time spent in an unpleasant
diagnostic experience would have great value to the patient. Since
the method greatly decreases the amount of time required for data
acquisition, it would, for example, greatly increase the
productivity of measurements involving very sensitive measurements
requiring significant signal averaging. One could envision, for
example, making sensitive measurements of a relatively short lived
free radical possible, whereas such measurements are now not
possible, by decreasing the required data acquisition time by
better than a factor of ten. The quite substantial increase in the
rate of data acquisition time will quite generally speed up any
application in which an analytical measurement such as continuous
wave magnetic resonance is utilized as a diagnostic tool in applied
research. For example, consider a research and development project
in which specific performance limiting defects are identified in a
semiconductor device technology. A production manager, made aware
of the specific performance limiting defects may alter processing
chemistry to ameliorate or eliminate problems caused by the
defects. Because our method provides a much more rapid means of
acquiring data, the process involved in the development of an
improved processing approach is quite significantly faster, saving
the organization involved significant research and development
costs.
[0041] Although we have described certain present preferred
embodiments of our method of signal processing of signals
containing repeatable measurements to remove noise, it should be
distinctly understood that the invention is not limited thereto but
may be variously embodied within the scope of the following
claims.
* * * * *